Calculation of the Stability of Phase Space Trajectories
using Molecular Dynamic Simulations

We derived an expression that explains why it is
so difficult to find initial microstates that will at long times,
under the influence of an external dissipative field and a thermostat,
lead to the Second Law violating nonequilibrium steady states.
To verify the derivation, nonequilibrium molecular dynamics simulations
of a system under the influence of shear were carried out. Computer
simulations were carried out to verify the predictions of nonlinear
response theory for classical systems subject to dissipative external
fields. The Kawasaki normalisation factor, used in the Kawasaki
nonlinear response expression, is shown to be unity. The influence
of Lyapunov instability on the lifetimes of antisteady states
was investigated using nonequilibrium molecular dynamics simulations.

What are the basic questions addressed?

Why, when a dissipative field is applied to a system
at equilibrium and the system is thermostatted, do we overwhelmingly
observe trajectories that satisfy the Second Law of Thermodynamics?
Can the predictions of nonlinear response theory be verified using
computer simulations? Does the Kawasaki normalisation factor
differ from unity? What factors contribute to the time for which
a trajectory can be reversed?

What are the results to date and future of the
work?

The expression relating the ratio of the probability
of observing the Second Law violating and obeying trajectories
was verified using molecular dynamics simulations of a system
under the influence of shear. Proof that the Kawasaki normalisation
factor is unity in an irreversible steady state relies on the
fact that the distribution of values of time averaged fluxes becomes
Gaussian as the length of time averaging becomes large, as expected
from the central limit theorem and a Green-Kubo type expression
exists for the standard deviation of the distribution. Simulations
were carried out which verified that these assumptions are valid.
For the first time, we were able to provide numerical evidence
that the predictions of nonlinear response theory agree with the
directly measured nonlinear response. The Kawasaki and the Transient
Time Correlation function expressions for nonlinear response were
found to be in agreement with the directly measured nonlinear
response for an autonomous system of just two particles.

The lifetime of antisteady states was found to be
inversely proportional to the smallest Lyapunov exponent of the
steady state system and proportional to the logarithm of the trajectory
error. These results were verified using nonequilibrium molecular
dynamics simulations. further studies of the stability of phase
space trajectories are currently being extended by investigation
of the Lyapunov exponents of model systems such as hard disks.

What computational techniques are used and why
is a supercomputer required?

Equilibrium and nonequilibrium molecular dynamics
simulations are used and are being developed. A supercomputer
is required to obtain statistically valid data for small systems
and due to large system size requirements.